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I want him to call me tomorrow.

In this sentence we see:

• 
  I, subject; want, verb; him, object.

What is the subject of to call? Him? But him is the object of the verb want. Is this a correct sentence? Or this better :

• I want that he calls me tomorrow?"

Thanks.

The answer to your general question differs from theory to theory. In context free grammar, ordinary transformational grammar, and probably in traditional grammar, it's no. A single sentence structure may not have a nominal which is simultaneously subject and object. In Relational Grammar, however, this is possible, and it might also be possible in McCawley's version of transformational grammar.

In ordinary transformational grammar, a NP can be subject of an embedded clause at one stage of derivation, then be moved into the higher clause by the transformation Subject Raising to Object. For instance, the "him" in "We believe him to be present" is the object of "believe" and is not a subject, yet at an earlier stage of derivation, it was the subject of "be present" (and not the object of "believe"). In this way, one can describe a construction in which a nominal is a grammatical object, but at the same time a logical subject (i.e., subject in deep structure).

The particular example you give is not clear cut, since the "him" might not be the grammatical object of "want". The passive form ?"He is wanted to call me tomorrow" is not fully acceptable. (But for the example I consider above, the passive "He is believed to be present" is fully acceptable.)

First, is it a grammatically correct sentence?  Yes.  This type of sentence is very common. 

As Greg Lee mentions, there are several different theories of English grammar, and different theories may label the parts of this sentence in different ways. 

In the framework that I use, "to call" is an infinitive and infinitives do not take subjects*.  However, infinitives and their phrases act as general modifiers. 

I want him to call me tomorrow. 

In this sentence, we see one clause.  The clause has the subject "I", the verb "want", the direct object "him", and the object complement "to call me tomorrow". 

We can compare this to an example which uses an adjective for its complement: 

I want him happy. 

Under this framework, "to call me tomorrow" has no more need of a subject than "happy".  Both relate to the object "him" in the same way -- as modifiers governed by the verb "want". 

 

Second, is the proposed substitution better?  No.  Grammatically, it might be just as good.  Idiomatically, it is unnatural. 

By "unnatural" I simply mean that, in my dialect, the sentence sounds foreign and awkward.  Another dialect may find that construction to be perfectly natural and unsurprising. 

 

Given a different governing verb, a complete clause as the direct object can certainly sound natural:

✓ I want him to call me tomorrow. 

✓ I hope that he calls me tomorrow. 

Swapping the verbs leads to sentences that sound unnatural: 

✗ I hope him to call me tomorrow. 

✗ I want that he calls me tomorrow. 

 

_______________ 

 

* That isn't to say that "to call" does not have an agent.  Semantic roles like agent and theme are separate and distinct from grammatical roles like subject and object.  That "him" is the object of "want" prevents it, in my framework, from being the subject of "to call".  At the same time, "him" represents both the theme of "want" and the agent of "to call".

I want him [to call me tomorrow].

Yes, him belongs syntactically in the matrix clause as object of want. But semantically, it belongs solely in the embedded clause as subject.

Him is not an argument of want. What I want is not him, but him to call me tomorrow. Syntactically, him is object of want, but semantically it relates only to the subordinate call clause (as understood subject), not to want. We thus say that him is a raised object because the verb it relates to is syntactically higher in the constituent structure than the one it relates to semantically.

There’s no need to recast your sentence; it’s fine both syntactically and semantically.

I want him to call me tomorrow is a grammatically perfect sentence. The sentence pattern goes like this;

1. Subject S ...... (I)
   


2. Verb V ........ (want)
   


3. Object O ......... (him)
   


4. Complement C ....... (to call)
   


5. Subject complement (optional) (me)
   


6. Here "tomorrow" is a NOUN which functions as time. Hence, an adjunct. A
   

In the above sentence to him is a nominal phrase, and this nominal phrase refers to a person whose name is implied.

Critically examine the sentence below;

" I want him(Peter) to call me tomorrow"

Now let us analyze the above sentence.

The him is representation of a masculine person we do not know. It functions as the object to the verb want.

However, the "him" is an OBJECTIVE PRONOUN. And objective pronouns like me, him, us, them her ( objective pronoun not that which shows possession, like her book...... ) do not function as Subject

Now to call is a to-infinitive verb, that is a particle "to" and a verb "call" which functions as object complement. Therefore objective pronouns do not occupy the Subject slot.

For example; one can not say,

• 
  Me like studying literature books.
  instead


• I like studying literature books.

Therefore, the objective pronoun him does not function as a Subject.

I want him to call me tomorrow

I is the subject, him is the object, and call is the verb, its simple really